Partitioning populations of real and manufactured particles using spherical harmonics representations

The objective of this paper is to review the discriminating power of spherical harmonic representations of the surfaces of particles, specifically for the task of partitioning a population of particles into clusters. By comparing the results of partitioning tasks on both classical shape descriptors and spherical harmonics, and by performing these tasks on both real populations of grains and populations of manufactured (analytical) grains, it is possible to come to some conclusions about the geometrical content of spherical harmonics. Spherical harmonics are compared to the classical shape descriptors of sphericity, convexity, corners, roundness, and the indices of elongation and flatness. Eight types of manufactured particles, each with various numbers of free parameters, are used to generate artificial populations that are to be partitioned. The types of manufactured particles are called spheres, prolates, oblates, cubes, cylinders, peanuts, diamonds and popcorn, but only the last two contain non-smooth surfaces either at points or along curves. Detailed experiments are performed to partition both manufactured and real populations using a Gaussian mixture model and the Bayesian information criterion to determine the number of such subpopulations. The experiments show that classical shape descriptors provide geometrical characterizations that can only separate out the most spherical particles from other particles, while spherical harmonics are effective throughout the range of smooth to non-smooth particles. Clustering is also performed for four different real populations of particles and contingency matrices are presented. The paper also includes several new benchmarks for the verification of calculations of spherical harmonics from micro-tomographic scans. • Generation of families of synthetic particles with known geometrical properties and subpopulations. • Comparison of the ability of Spherical Harmonics (SH) and Classical Shape Descriptors (CSD) to recover subpopulations of synthetic particles. • Discussion of the qualitative difference between clusters found by SH and CSD on real particles. • Analysis of various sources of errors arising from particle scan modeling, numerical integration, and SH convergence.